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I learned reading this question that $\mathrm{GL}(n,p)$ elements have at most a multiplicative order of $p^n -1$.

I would like to know how many matrices have an order of exactly $p^n -1$. Do they represent a majority of matrices in $\mathrm{GL}(n,p)$?

The answer suggested to build matrices of maximal order this way.

Consider a degree n monic polynomial $P_n$ whose root is a generator $ξ$ of $F^∗_{p^n}$. Then a matrix with $P_n$ as its characteristic polynomial has order at least $p^n−1$ since $ξ$ is its eigenvalue.

I don't know how to create a such matrix, yet enumerate them...

Thank you so much for any help.

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    $\begingroup$ Rational canonical form is very relevant here. This is really a question about linear algebra and possible minimum polynomials. $\endgroup$
    – Geoff Robinson
    Commented May 9, 2020 at 15:10
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    $\begingroup$ To be precise, I think the number of such matrices is $\frac{\phi(p^{n}-1)|{\rm GL}(n,p)|}{n(p^{n}-1)}$. $\endgroup$
    – Geoff Robinson
    Commented May 9, 2020 at 15:18
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    $\begingroup$ The way to see this is to see that there are $\frac{\phi(p^{n}-1)}{n}$ conjugacy classes of elements of order $p^{n}-1$, and then note that if $A$ is a matrix of multiplicative order $p^{n}-1$ in $G = {\rm GL}(n,p)$, then $|C_{G}(A)| = p^{n}-1.$ $\endgroup$
    – Geoff Robinson
    Commented May 9, 2020 at 15:24

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